A232546 - OEIS

%I #81 Apr 05 2017 08:00:45

%S 1,-18,54,108,486,2916,20412,157464,1299078,11258676,101328084,

%T 939587688,8926083036,86514343272,852784240824,8527842408240,

%U 86344404383430,883760374277460,9132190534200420,95167038198509640,999253901084351220,10563541240034570040

%N Expansion of (1 - 12*x)^(3/2) in powers of x.

%C From _Ralf Steiner_, Apr 04 2017: (Start)

%C By analytic continuation to the entire complex plane there exist regularized values for divergent sums such as:

%C Sum_{k>=0} a(k)^2/16^k = 2F1(-3/2,-3/2,1,9).

%C Sum_{k>=0} a(k) / 6^k = -i. (End)

%H G. C. Greubel, <a href="/A232546/b232546.txt">Table of n, a(n) for n = 0..930</a>

%H R. Steiner, <a href="https://www.researchgate.net/publication/315740800">Sums of OEIS-A232546</a>, ResearchGate, 2017. - _Ralf Steiner_, Apr 04 2017

%F 0 = a(n+2)*(a(n+1) - 42*a(n)) + 18*a(n+1)*(a(n+1) + 8*a(n)) for all n in Z.

%F a(n+2) = 54 * A000168(n). a(n) = 3^n * A002421(n). Convolution inverse of A115903.

%F a(n) = 6*(2*n-5)*a(n-1)/n. - _R. J. Mathar_, Nov 23 2014

%F G.f.: 1F0(-3/2;;12x). - _R. J. Mathar_, Aug 09 2015

%F For n>=2, a(n) = 4*3^(n+1)*(2*n-4)! / ((n-2)!*n!). - _Vaclav Kotesovec_, Apr 02 2017

%F Sum_{k>=0} a(k) / 12^k = 0. - _Ralf Steiner_, Apr 04 2017

%e G.f. = 1 - 18*x + 54*x^2 + 108*x^3 + 486*x^4 + 2916*x^5 + 20412*x^6 + ...

%t a[ n_] := SeriesCoefficient[ (1 - 12 x)^(3/2), {x, 0, n}];

%t Table[9/Sqrt[Pi] 12^n Gamma[-1/2 + n]/Gamma[2 + n], {n, -1, 20}] (* _Ralf Steiner_, Apr 01 2017 *)

%t Flatten[{1, -18, Table[4*3^(n+1)*(2*n-4)!/((n-2)!*n!), {n, 2, 25}]}] (* _Vaclav Kotesovec_, Apr 02 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( (1 - 12 * x + x * O(x^n))^(3/2), n))};

%Y Cf. A000168, A002421, A115903.

%K sign

%O 0,2

%A _Michael Somos_, Nov 25 2013